- Email: [email protected]
New representation of n-mode squeezed state gained via n-partite entangled state
Optics Communications 254 (2005) 256–261 www.elsevier.com/locate/optcom
New representation of n-mode squeezed state gained via n-partite entangled state q Nian-Quan Jiang Department of Physics, Wenzhou Normal College, Wenzhou 325027, China Received 30 January 2005; received in revised form 27 March 2005; accepted 25 May 2005
Abstract By virtue of the n-partite entangled state, we extend the way of Agarwal–SimonÕs presenting single-mode squeezed state to n-mode case and find a new representation of the n-mode squeezed state. This n-mode squeezed state is also an entangled state and can be a superposition of n-mode coherent states. 2005 Elsevier B.V. All rights reserved. PACS: 42.50.Dv; 03.67.a; 03.65.Bz Keywords: n-Mode squeezed state; Entangled state; New representation of squeezing
Squeezed states have been important topic since 1970s due to their wide applications in optical communication and precise measurement in quantum optics [1]. They exhibit interesting nonclassical behavior. Many attempts have been made to find new squeezed states and new form of squeezing operators so that new experimental implementation could be proposed [2,3]. For the single-mode squeezed state wave func1=4 tion Wðx; rx Þ ¼ ½1=ðpr2x Þ expðx2 =2r2x Þ (its wave packetÕs width is r2x =2) Agarwal and Simon found a new representation [4] 1 2 2 ðs jWix s1=2 exp 1ÞX ð1Þ s2x ¼ 1=r2x ; x 1 j0i 2 x 1=4
or for Wðp; rp Þ ¼ ½1=ðpr2p Þ expðp2 =2r2p Þ, 1 jWip sp1=2 exp ðs2p 1ÞP 21 j0i s2p ¼ 1=r2p ; 2 q
Work supported by the National Natural Science Foundation of China under Grant 10447128. E-mail address: [email protected]
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.05.046
ð2Þ
N.-Q. Jiang / Optics Communications 254 (2005) 256–261
where X1 and P1 are the coordinates and momentum operators, respectively. Let f ¼ 12 ðs2p 1Þ we have the single-mode squeezing operator, S 1 ð1 2f Þ
1=2 fP 2 1
e
1=2 fP 2 1
e
sp ¼ ek , ð3Þ
.
Using S1, Eq. (2) can be expressed as jWip ð1 2f Þ
257
j0i ¼ sech1=2 k exp
1 y2 a1 tanh k j0i; 2
ð4Þ
which is just the traditional standard form of the single-mode squeezed vacuum state. Moreover, Agarwal– SimonÕs representation shows that the single-mode squeezed state can be a superposition of coherent states on a line in phase space, i.e., !1=2 Z ! 1 sp x2 jWip ¼ dx exp 2 ð5Þ jxicoh ; pðs2p 1Þ sp 1 1 pffiffiffi where jxicoh ¼ expði 2P 1 xÞj0i is coherent state [5,6]. Enlightened by Agarwal–SimonÕs idea and employing the two-mode entangled state, in [7] we have derived a new form of the two-mode squeezing operator S 2 ð1 2f Þ
1=2 f ½ðX 1 X 2 Þ2 þðP 1 þP 2 Þ2 =2
;
ð6Þ
1 f y : exp a1 a2 a1 ay2 . 1f 1f
ð7Þ
e
whose normally product form is S 2 ¼ ð1 2f Þ
1=2
Operating S2 on the vacuum state |00æ will yield the two-mode squeezed state S 2 j00i ¼ sechk exp½ay1 ay2 tanh kj00i;
ð8Þ
which is just the traditional standard form of the two mode squeezed vacuum state. An interesting problem thus naturally arises: Can Agarwal–SimonÕs idea be extended to n-mode case? We will deal with this problem in the article. Firstly, we need introduce a n-mode entangled state with continuous variables. To derive compact result, we choose the Jacobian coordinates used in traditional many-body theory [8] as observable variables (One of the advantages of using Jacobian coordinates is that the motion of center-of-mass can be separated off from the relativeP motion of many particles). Noting that for n-partite system Pnthe Jacobian j1 1 coordinates operators X j j1 X ðj ¼ 2; . . . ; nÞ and the total momentum operator k k¼1 i¼1 P i are permutable with each other, we can introduce the common eigenstate of them [9] " pffiffiffi n n n X pffiffiffi X 1 1 1 2 2ip X 2 ðj 1Þvj p þ ayj þ 2 vj ayj jpv2 v3 . . . vn i ¼ pffiffiffi n=4 exp 2j 2n np n j¼2 j¼1 j¼1 ! # X j n n n X X pffiffiffi X vj 2 1 1 2 ayk þ ayj ayk þ ay2 j0 0i; j n n 2 j k¼1 j¼1 jk;j;k¼1 j¼1 where v1 = 0 which obeys the eigenvector equations ! n X P i jpv2 v3 . . . vn i ¼ pjpv2 v3 . . . vn i i¼1
and
ð9Þ
ð10Þ
258
N.-Q. Jiang / Optics Communications 254 (2005) 256–261
!
Xj
j1 1 X X k jpv2 v3 vn i ¼ vj jpv2 v3 . . . vn i j 1 k¼1
j ¼ 2; . . . ; n.
The Schmidt decomposition of |pv2v3. . .vnæ in n-mode coordinates basis is ! n X 1 vk jpv2 v3 . . . vn i ¼ pffiffiffiffiffiffi exp ip ; k 2p k¼2 Z 1 n1 X 1 v i expðixpÞ; dxjxi1 jx þ v2 i2 jx þ vn þ j j n 1 j¼2
ð11Þ
ð12Þ
where |xæi (i = 1,2,. . .,n) is the coordinates eigenstate of Xi, Xi|xæi = x|xæi 1=4
jxii ¼ p
1 2 pffiffiffi y 1 y2 exp x þ 2xai ai j0ii . 2 2
ð13Þ
According to the standard theory of Schmidt decomposition of an entangled state [10], we know that |pv2v3. . .vnæ is an entangled state. Using the technique of integral within an ordered product (IWOP) of operators [11] we can directly prove |pv2. . .vnæ span a complete set Z
1
dp dv2 . . . dvn jp; v2 ; . . . ; vn ihp; v2 ; . . . ; vn j ¼ 1.
ð14Þ
1
Specially, when n = 3, Eq. (9) reduces to " pffiffiffi pffiffiffi pffiffiffi 3 1 1 2 1 2 1 2 2 X 2 3 2 3 y y v2 þ v3 a1 þ v2 v3 ay2 ip jpv2 v3 i ¼ pffiffiffi exp v2 v3 p þ aj 4 3 6 3 3 2 3 2 3p3=4 j¼1 # pffiffiffi 3 3 2 2 2 X 1X v3 ay3 þ þ ayj ayk ay2 j0 0i; ð15Þ 3 3 jk;j;k¼1 6 j¼1 j which obeys ðP 1 þ P 2 þ P 3 Þjpv2 v3 i ¼ pjpv2 v3 i; ðX 2 X 1 Þjpv2 v3 i ¼ v2 jpv2 v3 i
ð16Þ ð17Þ
and
1 X 3 ðX 1 þ X 2 Þ jpv2 v3 i ¼ v3 jpv2 v3 i; 2
If we let v2 ! y 2 ; v3 ! 12 y 2 y 3 ; then |pv2v3æ becomes pffiffiffi 3 2 i 2p X 1 p 1 2 2 exp y 2 þ y 3 y 2 y 3 þ ay jpy 2 y 3 i ¼ pffiffiffi 3 i¼1 i 6 3 3p3=4 # pffiffiffi pffiffiffi 3 3 2X 1X 2y 2 y 2y 3 y y y y y y y y2 a1 2a2 þ a3 þ a1 þ a2 2a3 þ þ aa a j000i. 3 i
ð18Þ
ð19Þ
N.-Q. Jiang / Optics Communications 254 (2005) 256–261
Corresponding to (3) and (6), we introduce n-mode squeezing operator 8 2 !2 !2 39 j1 n n < X = X X j 1 1 1 n=4 Xj S n ð1 2f Þ exp f 4 Xk þ Pi 5 . : j¼2 j ; j 1 k¼1 n i¼1
259
ð20Þ
Using (10), (11) andP(14) as well as the normally ordered form of the vacuum projector j0 0ih0 0j ¼: exp½ ni¼1 ayi ai :, we can derive the normal product form of Sn, " !# Z 1 n X j1 2 1 2 n=4 vj þ p S n ¼ ð1 2f Þ dp dv2 . . . dvn exp f jp; v2 ; . . . ; vn ihp; v2 ; . . . ; vn j j n 1 j¼2 n=4 ð1 2f Þ f n K ¼ : exp þ K þ 2K ; ð21Þ þ 0 n=2 ð1 f Þ 2 ð1 f Þ where Kþ
n n 1X 2 2 X ð1 Þay2 ay ay ; 2 i¼1 n i n i
K
K yþ ; 2K 0
n X i¼1
ayi ai
ð22Þ
n þ ; 2
compose a closed SU(1,1) Lie algebra, ½K 0 ; K þ ¼ K þ ; ½K 0 ; K ¼ K ; ½K ; K þ ¼ 2K 0 . Operating (21) on |0 0æ we derive n=4
ð1 2f Þ f K þ j0 0i ¼ sechn=2 k exp½K þ tanh k 0 0i; S n j0 0i ¼ exp n=2 1f ð1 f Þ s2 1
ð23Þ
ð24Þ
2s
p where f ¼ 12 ð1 s2p Þ , sp = ek, tanh k ¼ sp2 þ1 ; sechk ¼ s2 þ1 , which is the standard form of the n-mode squeezed p p vacuum state. Calculating the quantum fluctuation of the two photon-field quadratures
3 3 1 X 1 X P i ; X qua ¼ pffiffiffi X i; P qua ¼ pffiffiffi n i¼1 n i¼1
ð25Þ
in the state Sn|0 0æ, we obtain 2 2 1 1 h 4X qua i ¼ s2p ; h 4P qua i ¼ s2 ; 4 4 p which show the standard squeezing. Thus, 8 2 !2 !2 39 j1 n n < X = X X j 1 1 1 n=4 Xj Xk þ P i 5 j0 0i; ð1 2f Þ exp f 4 : j¼2 j ; j 1 k¼1 n i¼1
ð26Þ
ð27Þ
is the new representation of n-mode squeezed vacuum state in the sense of Agarwal and Simon. From (27) we see that for a n-partite system once the Jacobian coordinates and total momentum are chosen as observable variables, n-mode squeezing operator is just the form similar to (3) which was introduced by Agarwal and Simon, so the Jacobian coordinates exhibit some symmetry of n-partite system. Especially, when n = 2 (20) reduce to (6); when n = 3 (20) becomes,
260
N.-Q. Jiang / Optics Communications 254 (2005) 256–261
8 2 !2 39 2 3 < 1 = X 2 1 1 3=4 2 X 3 ðX 1 þ X 2 Þ þ S 3 ð1 2f Þ exp f 4 ðX 2 X 1 Þ þ Pi 5 . : 2 ; 3 2 3 i¼1
ð28Þ
Moreover, the form (27) provides us a new view for the construction of the n-mode squeezed states, due to
"
# j1 i1 1 X 1 X X k; X i X k ¼ 0; Xj j 1 k¼1 i 1 k¼1 " # j1 n X 1 X X k; P i ¼ 0; i; j ¼ 2; 3; ; n; Xj j 1 k¼1 i¼1
ð29Þ
we have S n j0 0i ¼
nð1 2f Þn=4 n=2
ð4f pÞ "
Z
Z
1
1
du 1
dv2 . . . dvn 1 n X
!#
1 X n n1
n1 P
vn Xk 1 j 2 k¼1 nu2 þ vj euP ev2 ðX 2 X 1 Þ e j0 0i; ð30Þ 4f j1 j¼2 Pn1 1 where euP ev2 ðX 2 X 1 Þ evn ðX n n1 k¼1 X k Þ j0 0i is a n-mode coherent state. Eq. (30) indicates that the n-mode squeezed state can be a superposition of n-mode coherent states only along a line in the complex plane of each mode. In summary, we have extended the new representation of squeezed state in the sense of Agarwal and Simon to the n-mode case, in our discussion the n-partite entangled state |pv2. . .vnæ plays an essential role. The new representation implies that the n-mode squeezed state can also be a superposition of coherent states. Because the n-mode squeezed vacuum state Sn|0 0æ is also entangled state, they may have potential applications in quantum information.
exp
Acknowledgement Thank referees for their constructive comments.
References [1] See, e.g., G.M. DÕAriano, M.G. Rassetti, J. Katriel, A.I. Solomon, in: P. Tombesi, E.R. Pike (Eds.), Squeezed and Nonclassical Light, Plenum, New York, 1989, p. 301; V. Buzˇek, J. Mod. Opt. 37 (1990) 303; R. Loudon, P.L. Knight, J. Mod. Opt. 34 (1987) 709. [2] See, e.g., L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge, 1995. [3] V.V. Dodonov, J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1. [4] G.S. Agarwal, R. Simon, Opt. Commun. 92 (1992) 105. [5] R.J. Glauber, Phys. Rev. 130 (1963) 2529; R.J. Glauber, Phys. Rev. 131 (1963) 2766. [6] J.R. Klauder, B.S. Skargerstam, Coherent States, World Scientific, Singapore, 1985. [7] H.Y. Fan, N.Q. Jiang, H.L. Lu, Opt. Commun. 234 (2004) 277. [8] D.I. Blokhintsev, Quantum Mechanics, Reidel, Dordrecht, 1964. [9] N.-Q. Jiang, Phys. Lett. A 339 (2005) 255. [10] J. Preskill, Physics Lectures, California Institute of Technology, Pasadena, 1998, p. 229.
N.-Q. Jiang / Optics Communications 254 (2005) 256–261 [11] H.Y. Fan, H.R. Zaidi, J.R. Klauder, Phys. Rev. D 35 (1987) 1831; H.Y. Fan, J. Opt. B: Quantum Semiclass. Opt. 5 (2003) R147. [12] H.Y. Fan, N.Q. Jiang, J. Opt. B: Quantum Semiclass. Opt. 5 (2003) 283.
261
New representation of n-mode squeezed state gained via n-partite entangled state q Nian-Quan Jiang Department of Physics, Wenzhou Normal College, Wenzhou 325027, China Received 30 January 2005; received in revised form 27 March 2005; accepted 25 May 2005
Abstract By virtue of the n-partite entangled state, we extend the way of Agarwal–SimonÕs presenting single-mode squeezed state to n-mode case and find a new representation of the n-mode squeezed state. This n-mode squeezed state is also an entangled state and can be a superposition of n-mode coherent states. 2005 Elsevier B.V. All rights reserved. PACS: 42.50.Dv; 03.67.a; 03.65.Bz Keywords: n-Mode squeezed state; Entangled state; New representation of squeezing
Squeezed states have been important topic since 1970s due to their wide applications in optical communication and precise measurement in quantum optics [1]. They exhibit interesting nonclassical behavior. Many attempts have been made to find new squeezed states and new form of squeezing operators so that new experimental implementation could be proposed [2,3]. For the single-mode squeezed state wave func1=4 tion Wðx; rx Þ ¼ ½1=ðpr2x Þ expðx2 =2r2x Þ (its wave packetÕs width is r2x =2) Agarwal and Simon found a new representation [4] 1 2 2 ðs jWix s1=2 exp 1ÞX ð1Þ s2x ¼ 1=r2x ; x 1 j0i 2 x 1=4
or for Wðp; rp Þ ¼ ½1=ðpr2p Þ expðp2 =2r2p Þ, 1 jWip sp1=2 exp ðs2p 1ÞP 21 j0i s2p ¼ 1=r2p ; 2 q
Work supported by the National Natural Science Foundation of China under Grant 10447128. E-mail address: [email protected]
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.05.046
ð2Þ
N.-Q. Jiang / Optics Communications 254 (2005) 256–261
where X1 and P1 are the coordinates and momentum operators, respectively. Let f ¼ 12 ðs2p 1Þ we have the single-mode squeezing operator, S 1 ð1 2f Þ
1=2 fP 2 1
e
1=2 fP 2 1
e
sp ¼ ek , ð3Þ
.
Using S1, Eq. (2) can be expressed as jWip ð1 2f Þ
257
j0i ¼ sech1=2 k exp
1 y2 a1 tanh k j0i; 2
ð4Þ
which is just the traditional standard form of the single-mode squeezed vacuum state. Moreover, Agarwal– SimonÕs representation shows that the single-mode squeezed state can be a superposition of coherent states on a line in phase space, i.e., !1=2 Z ! 1 sp x2 jWip ¼ dx exp 2 ð5Þ jxicoh ; pðs2p 1Þ sp 1 1 pffiffiffi where jxicoh ¼ expði 2P 1 xÞj0i is coherent state [5,6]. Enlightened by Agarwal–SimonÕs idea and employing the two-mode entangled state, in [7] we have derived a new form of the two-mode squeezing operator S 2 ð1 2f Þ
1=2 f ½ðX 1 X 2 Þ2 þðP 1 þP 2 Þ2 =2
;
ð6Þ
1 f y : exp a1 a2 a1 ay2 . 1f 1f
ð7Þ
e
whose normally product form is S 2 ¼ ð1 2f Þ
1=2
Operating S2 on the vacuum state |00æ will yield the two-mode squeezed state S 2 j00i ¼ sechk exp½ay1 ay2 tanh kj00i;
ð8Þ
which is just the traditional standard form of the two mode squeezed vacuum state. An interesting problem thus naturally arises: Can Agarwal–SimonÕs idea be extended to n-mode case? We will deal with this problem in the article. Firstly, we need introduce a n-mode entangled state with continuous variables. To derive compact result, we choose the Jacobian coordinates used in traditional many-body theory [8] as observable variables (One of the advantages of using Jacobian coordinates is that the motion of center-of-mass can be separated off from the relativeP motion of many particles). Noting that for n-partite system Pnthe Jacobian j1 1 coordinates operators X j j1 X ðj ¼ 2; . . . ; nÞ and the total momentum operator k k¼1 i¼1 P i are permutable with each other, we can introduce the common eigenstate of them [9] " pffiffiffi n n n X pffiffiffi X 1 1 1 2 2ip X 2 ðj 1Þvj p þ ayj þ 2 vj ayj jpv2 v3 . . . vn i ¼ pffiffiffi n=4 exp 2j 2n np n j¼2 j¼1 j¼1 ! # X j n n n X X pffiffiffi X vj 2 1 1 2 ayk þ ayj ayk þ ay2 j0 0i; j n n 2 j k¼1 j¼1 jk;j;k¼1 j¼1 where v1 = 0 which obeys the eigenvector equations ! n X P i jpv2 v3 . . . vn i ¼ pjpv2 v3 . . . vn i i¼1
and
ð9Þ
ð10Þ
258
N.-Q. Jiang / Optics Communications 254 (2005) 256–261
!
Xj
j1 1 X X k jpv2 v3 vn i ¼ vj jpv2 v3 . . . vn i j 1 k¼1
j ¼ 2; . . . ; n.
The Schmidt decomposition of |pv2v3. . .vnæ in n-mode coordinates basis is ! n X 1 vk jpv2 v3 . . . vn i ¼ pffiffiffiffiffiffi exp ip ; k 2p k¼2 Z 1 n1 X 1 v i expðixpÞ; dxjxi1 jx þ v2 i2 jx þ vn þ j j n 1 j¼2
ð11Þ
ð12Þ
where |xæi (i = 1,2,. . .,n) is the coordinates eigenstate of Xi, Xi|xæi = x|xæi 1=4
jxii ¼ p
1 2 pffiffiffi y 1 y2 exp x þ 2xai ai j0ii . 2 2
ð13Þ
According to the standard theory of Schmidt decomposition of an entangled state [10], we know that |pv2v3. . .vnæ is an entangled state. Using the technique of integral within an ordered product (IWOP) of operators [11] we can directly prove |pv2. . .vnæ span a complete set Z
1
dp dv2 . . . dvn jp; v2 ; . . . ; vn ihp; v2 ; . . . ; vn j ¼ 1.
ð14Þ
1
Specially, when n = 3, Eq. (9) reduces to " pffiffiffi pffiffiffi pffiffiffi 3 1 1 2 1 2 1 2 2 X 2 3 2 3 y y v2 þ v3 a1 þ v2 v3 ay2 ip jpv2 v3 i ¼ pffiffiffi exp v2 v3 p þ aj 4 3 6 3 3 2 3 2 3p3=4 j¼1 # pffiffiffi 3 3 2 2 2 X 1X v3 ay3 þ þ ayj ayk ay2 j0 0i; ð15Þ 3 3 jk;j;k¼1 6 j¼1 j which obeys ðP 1 þ P 2 þ P 3 Þjpv2 v3 i ¼ pjpv2 v3 i; ðX 2 X 1 Þjpv2 v3 i ¼ v2 jpv2 v3 i
ð16Þ ð17Þ
and
1 X 3 ðX 1 þ X 2 Þ jpv2 v3 i ¼ v3 jpv2 v3 i; 2
If we let v2 ! y 2 ; v3 ! 12 y 2 y 3 ; then |pv2v3æ becomes pffiffiffi 3 2 i 2p X 1 p 1 2 2 exp y 2 þ y 3 y 2 y 3 þ ay jpy 2 y 3 i ¼ pffiffiffi 3 i¼1 i 6 3 3p3=4 # pffiffiffi pffiffiffi 3 3 2X 1X 2y 2 y 2y 3 y y y y y y y y2 a1 2a2 þ a3 þ a1 þ a2 2a3 þ þ aa a j000i. 3 i
ð18Þ
ð19Þ
N.-Q. Jiang / Optics Communications 254 (2005) 256–261
Corresponding to (3) and (6), we introduce n-mode squeezing operator 8 2 !2 !2 39 j1 n n < X = X X j 1 1 1 n=4 Xj S n ð1 2f Þ exp f 4 Xk þ Pi 5 . : j¼2 j ; j 1 k¼1 n i¼1
259
ð20Þ
Using (10), (11) andP(14) as well as the normally ordered form of the vacuum projector j0 0ih0 0j ¼: exp½ ni¼1 ayi ai :, we can derive the normal product form of Sn, " !# Z 1 n X j1 2 1 2 n=4 vj þ p S n ¼ ð1 2f Þ dp dv2 . . . dvn exp f jp; v2 ; . . . ; vn ihp; v2 ; . . . ; vn j j n 1 j¼2 n=4 ð1 2f Þ f n K ¼ : exp þ K þ 2K ; ð21Þ þ 0 n=2 ð1 f Þ 2 ð1 f Þ where Kþ
n n 1X 2 2 X ð1 Þay2 ay ay ; 2 i¼1 n i n i
K
K yþ ; 2K 0
n X i¼1
ayi ai
ð22Þ
n þ ; 2
compose a closed SU(1,1) Lie algebra, ½K 0 ; K þ ¼ K þ ; ½K 0 ; K ¼ K ; ½K ; K þ ¼ 2K 0 . Operating (21) on |0 0æ we derive n=4
ð1 2f Þ f K þ j0 0i ¼ sechn=2 k exp½K þ tanh k 0 0i; S n j0 0i ¼ exp n=2 1f ð1 f Þ s2 1
ð23Þ
ð24Þ
2s
p where f ¼ 12 ð1 s2p Þ , sp = ek, tanh k ¼ sp2 þ1 ; sechk ¼ s2 þ1 , which is the standard form of the n-mode squeezed p p vacuum state. Calculating the quantum fluctuation of the two photon-field quadratures
3 3 1 X 1 X P i ; X qua ¼ pffiffiffi X i; P qua ¼ pffiffiffi n i¼1 n i¼1
ð25Þ
in the state Sn|0 0æ, we obtain 2 2 1 1 h 4X qua i ¼ s2p ; h 4P qua i ¼ s2 ; 4 4 p which show the standard squeezing. Thus, 8 2 !2 !2 39 j1 n n < X = X X j 1 1 1 n=4 Xj Xk þ P i 5 j0 0i; ð1 2f Þ exp f 4 : j¼2 j ; j 1 k¼1 n i¼1
ð26Þ
ð27Þ
is the new representation of n-mode squeezed vacuum state in the sense of Agarwal and Simon. From (27) we see that for a n-partite system once the Jacobian coordinates and total momentum are chosen as observable variables, n-mode squeezing operator is just the form similar to (3) which was introduced by Agarwal and Simon, so the Jacobian coordinates exhibit some symmetry of n-partite system. Especially, when n = 2 (20) reduce to (6); when n = 3 (20) becomes,
260
N.-Q. Jiang / Optics Communications 254 (2005) 256–261
8 2 !2 39 2 3 < 1 = X 2 1 1 3=4 2 X 3 ðX 1 þ X 2 Þ þ S 3 ð1 2f Þ exp f 4 ðX 2 X 1 Þ þ Pi 5 . : 2 ; 3 2 3 i¼1
ð28Þ
Moreover, the form (27) provides us a new view for the construction of the n-mode squeezed states, due to
"
# j1 i1 1 X 1 X X k; X i X k ¼ 0; Xj j 1 k¼1 i 1 k¼1 " # j1 n X 1 X X k; P i ¼ 0; i; j ¼ 2; 3; ; n; Xj j 1 k¼1 i¼1
ð29Þ
we have S n j0 0i ¼
nð1 2f Þn=4 n=2
ð4f pÞ "
Z
Z
1
1
du 1
dv2 . . . dvn 1 n X
!#
1 X n n1
n1 P
vn Xk 1 j 2 k¼1 nu2 þ vj euP ev2 ðX 2 X 1 Þ e j0 0i; ð30Þ 4f j1 j¼2 Pn1 1 where euP ev2 ðX 2 X 1 Þ evn ðX n n1 k¼1 X k Þ j0 0i is a n-mode coherent state. Eq. (30) indicates that the n-mode squeezed state can be a superposition of n-mode coherent states only along a line in the complex plane of each mode. In summary, we have extended the new representation of squeezed state in the sense of Agarwal and Simon to the n-mode case, in our discussion the n-partite entangled state |pv2. . .vnæ plays an essential role. The new representation implies that the n-mode squeezed state can also be a superposition of coherent states. Because the n-mode squeezed vacuum state Sn|0 0æ is also entangled state, they may have potential applications in quantum information.
exp
Acknowledgement Thank referees for their constructive comments.
References [1] See, e.g., G.M. DÕAriano, M.G. Rassetti, J. Katriel, A.I. Solomon, in: P. Tombesi, E.R. Pike (Eds.), Squeezed and Nonclassical Light, Plenum, New York, 1989, p. 301; V. Buzˇek, J. Mod. Opt. 37 (1990) 303; R. Loudon, P.L. Knight, J. Mod. Opt. 34 (1987) 709. [2] See, e.g., L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge, 1995. [3] V.V. Dodonov, J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1. [4] G.S. Agarwal, R. Simon, Opt. Commun. 92 (1992) 105. [5] R.J. Glauber, Phys. Rev. 130 (1963) 2529; R.J. Glauber, Phys. Rev. 131 (1963) 2766. [6] J.R. Klauder, B.S. Skargerstam, Coherent States, World Scientific, Singapore, 1985. [7] H.Y. Fan, N.Q. Jiang, H.L. Lu, Opt. Commun. 234 (2004) 277. [8] D.I. Blokhintsev, Quantum Mechanics, Reidel, Dordrecht, 1964. [9] N.-Q. Jiang, Phys. Lett. A 339 (2005) 255. [10] J. Preskill, Physics Lectures, California Institute of Technology, Pasadena, 1998, p. 229.
N.-Q. Jiang / Optics Communications 254 (2005) 256–261 [11] H.Y. Fan, H.R. Zaidi, J.R. Klauder, Phys. Rev. D 35 (1987) 1831; H.Y. Fan, J. Opt. B: Quantum Semiclass. Opt. 5 (2003) R147. [12] H.Y. Fan, N.Q. Jiang, J. Opt. B: Quantum Semiclass. Opt. 5 (2003) 283.
261